# P3114: Kashaev volume conjecture for hyperbolic knots

- ID: `P3114`
- Reference: `kashaev-volume-conjecture`
- Page: https://theoremdb.org/statements/P3114
- Record maturity: Reviewed problem with recorded work

## Problem

For every hyperbolic knot \(K\subset S^3\), does \(\lim_{N\to\infty}(2\pi/N)\log|\langle K\rangle_N|=\operatorname{Vol}(S^3\setminus K)\), where \(\langle K\rangle_N\) is Kashaev's \(N\)-th quantum invariant?

### Context

Known frontier: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements.

Open boundary: A proof or counterexample for arbitrary hyperbolic knots remains unknown.

### Problem setup

- **Definition (hyperbolic knot).** A knot whose complement admits a complete finite-volume hyperbolic metric.
- **Definition (Kashaev invariant).** The Nth root-of-unity knot invariant introduced from the quantum dilogarithm.
- **Remark.** The conjecture equates exponential growth of quantum knot invariants at roots of unity with the hyperbolic volume of the knot complement. The Kashaev formulation avoids normalization ambiguity in colored Jones notation.

### What counts as a solution

- Prove existence of the limit and the stated equality for every hyperbolic knot.
- Or give a hyperbolic knot for which the limit fails to exist or differs from hyperbolic volume.

## Status

OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. Exact unresolved remainder: A proof or counterexample for arbitrary hyperbolic knots remains unknown. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (Current status and exact unresolved remainder).** OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. Exact unresolved remainder: A proof or counterexample for arbitrary hyperbolic knots remains unknown.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements.

The exact unresolved remainder is: A proof or counterexample for arbitrary hyperbolic knots remains unknown.

A complete resolution must meet the following acceptance conditions:
- Prove existence of the limit and the stated equality for every hyperbolic knot.
- Or give a hyperbolic knot for which the limit fails to exist or differs from hyperbolic volume.

### Background and intake notes

- Original intake status: OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. Exact unresolved remainder: A proof or counterexample for arbitrary hyperbolic knots remains unknown.
- The release review checked 2 structured sources on 2026-08-01.
- Equivalent-formulation queries: Kashaev volume conjecture general hyperbolic knots open 2026; volume conjecture proven classes knots current status
- Strongest checked neighboring result: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements.
- Exact unresolved remainder: A proof or counterexample for arbitrary hyperbolic knots remains unknown.

### Other known results

- **Claim 2** (supported): The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. Unresolved remainder: A proof or counterexample for arbitrary hyperbolic knots remains unknown. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): A proof or counterexample for arbitrary hyperbolic knots remains unknown.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `kashaev-volume-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>R. M. KASHAEV, “The Hyperbolic Volume of Knots from the Quantum Dilogarithm”. Letters in Mathematical Physics 39(3) (1997), 269-275. DOI 10.1023/A:1007364912784. volume-growth conjecture https://doi.org/10.1023/A:1007364912784
   - Also cited at R. Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Letters in Mathematical Physics 39 (1997). volume-growth conjecture
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Introduces the invariant and the volume asymptotic.
   - Source used to assess the problem's recorded status.
   - For Kashaev volume conjecture for hyperbolic knots: This is the dated publication status for the canonical target Kashaev volume conjecture for hyperbolic knots.
   - Source named by the research packet.
2. <a id="reference-2"></a>Hitoshi Murakami and Jun Murakami, “The colored Jones polynomials and the simplicial volume of a knot”. Acta Mathematica 186(1) (2001), 85-104. DOI 10.1007/BF02392716. Conjecture 2.2 and the examples that follow https://doi.org/10.1007/BF02392716
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Identifies Kashaev's invariant with a specialization of the colored Jones polynomial and states the volume conjecture in that form.
   - Source used to assess the problem's recorded status.
   - For Kashaev volume conjecture for hyperbolic knots: Identifies Kashaev's invariant with a specialization of the colored Jones polynomial and states the volume conjecture in that form.